On complements and the factorization problem for Hopf algebras

Mathematical Research Letters, 1998

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH) [Ka]. It was proved that the conjecture is true for H of dimension p n , p prime [MW], and that if H has a 2−dimensional representation then dimH is even [NR]. In this paper we first prove in Theorem 1.4 that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H) = (dimH) 2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use it to prove in Theorem 1.5 that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily in Theorem 1.7 that Kaplansky's conjecture [Ka] on prime dimensional Hopf algebras over k is true by passing to their Drinfeld doubles (compare with [Z]). Second, we use a theorem of Deligne [De] to prove in Theorem 2.2 that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras [Dr2].

2010

We survey the results obtained so far in the study of semisimple Hopf algebras. The role of Kaplansky's sixth conjecture to the development of this study is presented in detail. The connection between fusion categories and this study is also emphasized. In the final Section we present a survey on the relation between Hopf algebra extensions and finite depth theory.

Journal of Algebra, 2001

We show that if a finite dimensional Hopf algebra H over ‫ރ‬ has a basis with respect to which all the structure constants are nonnegative, then H is isomorphic to the bi-cross-product Hopf algebra constructed by Takeuchi and Majid from a finite group G and a unique factorization G s G G of G into two subgroups.

Canadian Mathematical Bulletin, 1984

In this note we generalise a result of D. Husemoller to certain bipolynomial Hopf algebras and are able to give Hopf algebra decompositions for these. As an easy consequence of our approach we give a simplified derivation of recent results of P. Hoffman on polynomial generators for these algebras; we also give explicit systems of “Borel generators” for a related family of quotient Hopf algebras considered by Hoffman.

International Mathematical Forum, 2012

Let X = GM be a factorization of a group into a subgroup G and a subsemigroup M with identity and a left inverse property. A bialgebra H = kM k(G) with basis m ⊗ δ g where m ∈ M and g ∈ G is called a left Hopf algebra if there is a one-sided antipode map S such that S(m⊗ δ g) = (m ¡ g) L ⊗ δ (m£g) −1. In this paper, we show that the quantum double D(kM k(G)) can be generated by H = kM k(G) and its dual H * = k(M) kG with specific cross relations. Moreover, an interesting example for these left Hopf algebras is introduced.

2016

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra H, one can associate it to a Hopf algebroid Hpar which has the universal property that each partial representation of H can be factorized by an algebra morphism from Hpar. We define also the category of partial modules over a Hopf algebra H, which is the category of modules over its associated Hopf algebroid Hpar. The Hopf algebroid structure of Hpar enables us to enhance the category of partial H modules with a monoidal structure and such that the algebra objects in this category are the usual partial actions. Some examples of categories of partial H modules are explored. In particular we can describe fully the category of partially Z 2-graded modules. Contents 1. Introduction 1 2. Partial representations of groups 4 3. Partial representations of Hopf algebras 8 4. The universal partial "Hopf" algebra H par 15 4.1. H par and the universal property 15 4.2. H par as a partial smash product 18 4.3. H par as a Hopf algebroid 20 4.4. The subalgebra A of H par 25 5. The category of partial modules 28 6. Partial G-Gradings 33 6.1. Partially graded vector spaces and partially graded algebras 34 6.2. Partial Z 2 gradings 39 Acknowledgments 41 References 41

Journal of Algebra, 2015

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra H, one can associate it to a Hopf algebroid Hpar which has the universal property that each partial representation of H can be factorized by an algebra morphism from Hpar. We define also the category of partial modules over a Hopf algebra H, which is the category of modules over its associated Hopf algebroid Hpar. The Hopf algebroid structure of Hpar enables us to enhance the category of partial H modules with a monoidal structure and such that the algebra objects in this category are the usual partial actions. Some examples of categories of partial H modules are explored. In particular we can describe fully the category of partially Z 2-graded modules. Contents 1. Introduction 1 2. Partial representations of groups 4 3. Partial representations of Hopf algebras 8 4. The universal partial "Hopf" algebra H par 15 4.1. H par and the universal property 15 4.2. H par as a partial smash product 18 4.3. H par as a Hopf algebroid 20 4.4. The subalgebra A of H par 25 5. The category of partial modules 28 6. Partial G-Gradings 33 6.1. Partially graded vector spaces and partially graded algebras 34 6.2. Partial Z 2 gradings 39 Acknowledgments 41 References 41

2016

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in [2]. Brauer's theorem for kernels over group algebras is generalized to Hopf algebras.

Arxiv preprint math/9812118, 1998

Proceedings of the Edinburgh Mathematical Society, 2020

We consider the adjoint representation of a Hopf algebra H focusing on the locally finite part, Had fin, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative H (i.e., H is finitely generated as module over a cocommutative Hopf subalgebra), we show that Had fin is a Hopf subalgebra of H. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, Had fin is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.